Properties

Label 4002.2.a.bh
Level 4002
Weight 2
Character orbit 4002.a
Self dual Yes
Analytic conductor 31.956
Analytic rank 0
Dimension 7
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{3} q^{5} \) \(+ q^{6}\) \( -\beta_{6} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{3}\) \(+ q^{4}\) \( + \beta_{3} q^{5} \) \(+ q^{6}\) \( -\beta_{6} q^{7} \) \(+ q^{8}\) \(+ q^{9}\) \( + \beta_{3} q^{10} \) \( + ( 2 + \beta_{5} ) q^{11} \) \(+ q^{12}\) \( -\beta_{4} q^{13} \) \( -\beta_{6} q^{14} \) \( + \beta_{3} q^{15} \) \(+ q^{16}\) \( + ( 2 + \beta_{3} - \beta_{5} ) q^{17} \) \(+ q^{18}\) \( + ( 1 - \beta_{1} - \beta_{6} ) q^{19} \) \( + \beta_{3} q^{20} \) \( -\beta_{6} q^{21} \) \( + ( 2 + \beta_{5} ) q^{22} \) \(+ q^{23}\) \(+ q^{24}\) \( + ( 1 + \beta_{4} + 2 \beta_{6} ) q^{25} \) \( -\beta_{4} q^{26} \) \(+ q^{27}\) \( -\beta_{6} q^{28} \) \(- q^{29}\) \( + \beta_{3} q^{30} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{31} \) \(+ q^{32}\) \( + ( 2 + \beta_{5} ) q^{33} \) \( + ( 2 + \beta_{3} - \beta_{5} ) q^{34} \) \( + ( \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{35} \) \(+ q^{36}\) \( + ( -2 \beta_{2} - \beta_{5} ) q^{37} \) \( + ( 1 - \beta_{1} - \beta_{6} ) q^{38} \) \( -\beta_{4} q^{39} \) \( + \beta_{3} q^{40} \) \( + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{41} \) \( -\beta_{6} q^{42} \) \( + ( 3 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{43} \) \( + ( 2 + \beta_{5} ) q^{44} \) \( + \beta_{3} q^{45} \) \(+ q^{46}\) \( + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{47} \) \(+ q^{48}\) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{49} \) \( + ( 1 + \beta_{4} + 2 \beta_{6} ) q^{50} \) \( + ( 2 + \beta_{3} - \beta_{5} ) q^{51} \) \( -\beta_{4} q^{52} \) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{5} ) q^{53} \) \(+ q^{54}\) \( + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{55} \) \( -\beta_{6} q^{56} \) \( + ( 1 - \beta_{1} - \beta_{6} ) q^{57} \) \(- q^{58}\) \( + ( -2 - 2 \beta_{3} - \beta_{4} ) q^{59} \) \( + \beta_{3} q^{60} \) \( + ( 2 \beta_{3} - \beta_{5} ) q^{61} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{62} \) \( -\beta_{6} q^{63} \) \(+ q^{64}\) \( + ( 1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{65} \) \( + ( 2 + \beta_{5} ) q^{66} \) \( + ( 2 - \beta_{3} + 2 \beta_{4} ) q^{67} \) \( + ( 2 + \beta_{3} - \beta_{5} ) q^{68} \) \(+ q^{69}\) \( + ( \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{70} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{71} \) \(+ q^{72}\) \( + ( 2 + \beta_{2} - \beta_{4} + \beta_{6} ) q^{73} \) \( + ( -2 \beta_{2} - \beta_{5} ) q^{74} \) \( + ( 1 + \beta_{4} + 2 \beta_{6} ) q^{75} \) \( + ( 1 - \beta_{1} - \beta_{6} ) q^{76} \) \( + ( \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{77} \) \( -\beta_{4} q^{78} \) \( + ( -2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{79} \) \( + \beta_{3} q^{80} \) \(+ q^{81}\) \( + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{82} \) \( + ( 3 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{83} \) \( -\beta_{6} q^{84} \) \( + ( 4 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} ) q^{85} \) \( + ( 3 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{86} \) \(- q^{87}\) \( + ( 2 + \beta_{5} ) q^{88} \) \( + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} ) q^{89} \) \( + \beta_{3} q^{90} \) \( + ( \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{91} \) \(+ q^{92}\) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{93} \) \( + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{94} \) \( + ( -2 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{6} ) q^{95} \) \(+ q^{96}\) \( + ( -2 + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{97} \) \( + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{98} \) \( + ( 2 + \beta_{5} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(7q \) \(\mathstrut +\mathstrut 7q^{2} \) \(\mathstrut +\mathstrut 7q^{3} \) \(\mathstrut +\mathstrut 7q^{4} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut +\mathstrut 7q^{6} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 7q^{8} \) \(\mathstrut +\mathstrut 7q^{9} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut +\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut 7q^{12} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut +\mathstrut 7q^{18} \) \(\mathstrut +\mathstrut 6q^{19} \) \(\mathstrut +\mathstrut q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut +\mathstrut 11q^{22} \) \(\mathstrut +\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 7q^{24} \) \(\mathstrut +\mathstrut 12q^{25} \) \(\mathstrut -\mathstrut q^{26} \) \(\mathstrut +\mathstrut 7q^{27} \) \(\mathstrut -\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 7q^{29} \) \(\mathstrut +\mathstrut q^{30} \) \(\mathstrut +\mathstrut 3q^{31} \) \(\mathstrut +\mathstrut 7q^{32} \) \(\mathstrut +\mathstrut 11q^{33} \) \(\mathstrut +\mathstrut 18q^{34} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 7q^{36} \) \(\mathstrut +\mathstrut 5q^{37} \) \(\mathstrut +\mathstrut 6q^{38} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut +\mathstrut 25q^{41} \) \(\mathstrut -\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 20q^{43} \) \(\mathstrut +\mathstrut 11q^{44} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 7q^{46} \) \(\mathstrut +\mathstrut 7q^{48} \) \(\mathstrut +\mathstrut 7q^{49} \) \(\mathstrut +\mathstrut 12q^{50} \) \(\mathstrut +\mathstrut 18q^{51} \) \(\mathstrut -\mathstrut q^{52} \) \(\mathstrut -\mathstrut 4q^{53} \) \(\mathstrut +\mathstrut 7q^{54} \) \(\mathstrut +\mathstrut 17q^{55} \) \(\mathstrut -\mathstrut 2q^{56} \) \(\mathstrut +\mathstrut 6q^{57} \) \(\mathstrut -\mathstrut 7q^{58} \) \(\mathstrut -\mathstrut 17q^{59} \) \(\mathstrut +\mathstrut q^{60} \) \(\mathstrut +\mathstrut 5q^{61} \) \(\mathstrut +\mathstrut 3q^{62} \) \(\mathstrut -\mathstrut 2q^{63} \) \(\mathstrut +\mathstrut 7q^{64} \) \(\mathstrut +\mathstrut 7q^{65} \) \(\mathstrut +\mathstrut 11q^{66} \) \(\mathstrut +\mathstrut 15q^{67} \) \(\mathstrut +\mathstrut 18q^{68} \) \(\mathstrut +\mathstrut 7q^{69} \) \(\mathstrut -\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 15q^{71} \) \(\mathstrut +\mathstrut 7q^{72} \) \(\mathstrut +\mathstrut 14q^{73} \) \(\mathstrut +\mathstrut 5q^{74} \) \(\mathstrut +\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 6q^{76} \) \(\mathstrut -\mathstrut 12q^{77} \) \(\mathstrut -\mathstrut q^{78} \) \(\mathstrut -\mathstrut 4q^{79} \) \(\mathstrut +\mathstrut q^{80} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut +\mathstrut 25q^{82} \) \(\mathstrut +\mathstrut 14q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut +\mathstrut 34q^{85} \) \(\mathstrut +\mathstrut 20q^{86} \) \(\mathstrut -\mathstrut 7q^{87} \) \(\mathstrut +\mathstrut 11q^{88} \) \(\mathstrut +\mathstrut 8q^{89} \) \(\mathstrut +\mathstrut q^{90} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 7q^{92} \) \(\mathstrut +\mathstrut 3q^{93} \) \(\mathstrut -\mathstrut 18q^{95} \) \(\mathstrut +\mathstrut 7q^{96} \) \(\mathstrut -\mathstrut 18q^{97} \) \(\mathstrut +\mathstrut 7q^{98} \) \(\mathstrut +\mathstrut 11q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(16\) \(x^{5}\mathstrut +\mathstrut \) \(51\) \(x^{4}\mathstrut +\mathstrut \) \(45\) \(x^{3}\mathstrut -\mathstrut \) \(152\) \(x^{2}\mathstrut -\mathstrut \) \(54\) \(x\mathstrut +\mathstrut \) \(70\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{6} + \nu^{5} + 100 \nu^{4} - 18 \nu^{3} - 499 \nu^{2} - 18 \nu + 383 \)\()/43\)
\(\beta_{3}\)\(=\)\((\)\( 15 \nu^{6} - 3 \nu^{5} - 257 \nu^{4} + 54 \nu^{3} + 981 \nu^{2} + 269 \nu - 547 \)\()/43\)
\(\beta_{4}\)\(=\)\((\)\( -28 \nu^{6} - 3 \nu^{5} + 474 \nu^{4} + 11 \nu^{3} - 1771 \nu^{2} - 720 \nu + 743 \)\()/43\)
\(\beta_{5}\)\(=\)\((\)\( 40 \nu^{6} - 8 \nu^{5} - 671 \nu^{4} + 187 \nu^{3} + 2444 \nu^{2} + 359 \nu - 1215 \)\()/43\)
\(\beta_{6}\)\(=\)\((\)\( 52 \nu^{6} - 19 \nu^{5} - 868 \nu^{4} + 342 \nu^{3} + 3031 \nu^{2} + 428 \nu - 1214 \)\()/43\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1}\mathstrut +\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\)\(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(12\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(12\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(38\) \(\beta_{3}\mathstrut +\mathstrut \) \(18\) \(\beta_{2}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(111\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{6}\mathstrut +\mathstrut \) \(25\) \(\beta_{5}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut -\mathstrut \) \(101\) \(\beta_{3}\mathstrut -\mathstrut \) \(31\) \(\beta_{2}\mathstrut +\mathstrut \) \(108\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\)\()/2\)
\(\nu^{6}\)\(=\)\(-\)\(70\) \(\beta_{6}\mathstrut +\mathstrut \) \(69\) \(\beta_{5}\mathstrut +\mathstrut \) \(69\) \(\beta_{4}\mathstrut +\mathstrut \) \(231\) \(\beta_{3}\mathstrut +\mathstrut \) \(122\) \(\beta_{2}\mathstrut -\mathstrut \) \(59\) \(\beta_{1}\mathstrut +\mathstrut \) \(574\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.924948
0.586166
−1.56994
3.08877
2.76200
−3.61135
2.66930
1.00000 1.00000 1.00000 −4.09187 1.00000 −2.85370 1.00000 1.00000 −4.09187
1.2 1.00000 1.00000 1.00000 −1.65866 1.00000 −1.05824 1.00000 1.00000 −1.65866
1.3 1.00000 1.00000 1.00000 −1.59086 1.00000 1.20696 1.00000 1.00000 −1.59086
1.4 1.00000 1.00000 1.00000 0.566335 1.00000 2.05767 1.00000 1.00000 0.566335
1.5 1.00000 1.00000 1.00000 0.888451 1.00000 4.32361 1.00000 1.00000 0.888451
1.6 1.00000 1.00000 1.00000 3.16777 1.00000 −1.07153 1.00000 1.00000 3.16777
1.7 1.00000 1.00000 1.00000 3.71883 1.00000 −4.60478 1.00000 1.00000 3.71883
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{7} \) \(\mathstrut -\mathstrut T_{5}^{6} \) \(\mathstrut -\mathstrut 23 T_{5}^{5} \) \(\mathstrut +\mathstrut 21 T_{5}^{4} \) \(\mathstrut +\mathstrut 120 T_{5}^{3} \) \(\mathstrut -\mathstrut 44 T_{5}^{2} \) \(\mathstrut -\mathstrut 128 T_{5} \) \(\mathstrut +\mathstrut 64 \)
\(T_{7}^{7} \) \(\mathstrut +\mathstrut 2 T_{7}^{6} \) \(\mathstrut -\mathstrut 26 T_{7}^{5} \) \(\mathstrut -\mathstrut 44 T_{7}^{4} \) \(\mathstrut +\mathstrut 136 T_{7}^{3} \) \(\mathstrut +\mathstrut 168 T_{7}^{2} \) \(\mathstrut -\mathstrut 144 T_{7} \) \(\mathstrut -\mathstrut 160 \)
\(T_{11}^{7} \) \(\mathstrut -\mathstrut 11 T_{11}^{6} \) \(\mathstrut +\mathstrut 17 T_{11}^{5} \) \(\mathstrut +\mathstrut 169 T_{11}^{4} \) \(\mathstrut -\mathstrut 506 T_{11}^{3} \) \(\mathstrut -\mathstrut 516 T_{11}^{2} \) \(\mathstrut +\mathstrut 2432 T_{11} \) \(\mathstrut -\mathstrut 1024 \)